Exploration
The dynamic figure below follows up on the Matric Exam Geometry Problem - 1949, and will help you find how to determine the maximum of ∠BEC. It shows a circle constructed with its centre I on the perpendicular bisector of BC, and tangent to the line segment AD at G.
1) Drag I till the circle passes through B and C.
2) Next drag E to coincide with G while carefully watching what happens to ∠BEC. What do you notice?
3) Change the shape of the trapezium ABCD, repeat the preceding process, and check your observation again.
Challenge: Can you explain (prove) why the maximum of the angle has to be located at the position of E you identified?
4) Can you figure out a way via construction, not using experimentation and dragging as with this sketch, to precisely locate where the centre I of the circle should be located to determine a maximum for ∠BEC?
Maximizing the Angle
Hints
5) Try using the exterior angle theorem to prove why the maximum of angle BEC is located where it is.
6) Consider the locus of points equidistant from vertex B and line AD and/or the locus of points equidistant from vertex C and line AD.
7) Carefully reflect on your solution/proof. Is it necessary that AB // CD? Can you generalize further?
8) Click on Show Solution button to display a constructed solution.
Published Paper
Check your explorations above, by reading my 2015 paper in At Right Angles and Learning and Teaching Mathematics at Flashback to the Past: a 1949 Matric Geometry Question.
Earlier References
I'm grateful to Vladimir Dubrovsky, Moscow, Russia, who recently (Jan 2024) on Facebook pointed out that the maximum angle problem above appeared earlier in the so-called "Bus Problem" of Chapter 5 of the following publication: Vasilyev, N.B. & Gutenmacher, V.L. (1980). Straight Lines & Curves. Mir Publishers, Moscow.
The problem also directly relates to maximizing the place kicking angle in rugby (De Villiers, 1999).
Related Papers
De Villiers, M. (1998). Exploring loci on Sketchpad. Pythagoras, 47, Dec., pp. 71-73.
De Villiers, M. (1999). Place Kicking Locus in Rugby. Pythagoras, 49, Aug., pp. 64-67.
Some Related Links
Matric Exam Geometry Problem - 1949
All parabola are similar - i.e. have the same shape
Cyclic Hexagon Alternate Angles Sum Theorem
A generalization of the Cyclic Quadrilateral Angle Sum theorem
Angle Divider Theorem for a Cyclic Quadrilateral
Semi-regular Angle-gons and Side-gons: Generalizations of rectangles and rhombi
Alternate sides cyclic-2n-gons and Alternate angles circum-2n-gons
Nine-point centre (anticentre or Euler centre) & Maltitudes of Cyclic Quadrilateral
An extension of the IMO 2014 Problem 4
Back to "Dynamic Geometry Sketches"
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Created by Michael de Villiers, 14 August 2015 with WebSketchpad; updated 19/27/28 Jan 2024.